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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0. Updated on: 14 Aug 2019, 02:11
Originally posted by CareerGeek on 05 Aug 2019, 11:34.
Last edited by Bunuel on 14 Aug 2019, 02:11, edited 2 times in total.
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C
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61% (01:34) correct 39% (01:29) wrong based on 687 sessions

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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0.

A. 0
B. 1
C. 2
D. 3
E. 4
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C
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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0. 05 Aug 2019, 11:51
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x^2 -2|x| -8 =0

If x is 0 or positive, then x^2 -2x -8 =0
--> (x-4)(x+2)=0
--> x=4 (Yes) or x=-2 (No!)

If x is negative, then x^2 +2x -8 =0
--> (x+4)(x-2)=0
--> x=-4 (Yes) or x=2 (No!)

There are only two solutions:
x=-4 and x=4


Answer is (C)
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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0. 14 Aug 2019, 02:08
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Hey, why are -2 and +2 not considered a solution for the problem?
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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0. 14 Aug 2019, 02:13
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anuradha619
Hey, why are -2 and +2 not considered a solution for the problem?
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Because -2 and 2 do not satisfy x^2 - 2|x| - 8 = 0.

If x = -2, then x^2 - 2|x| - 8 = 4 - 4 - 8 = -8, not 0.
If x = 2, then x^2 - 2|x| - 8 = 4 - 4 - 8 = -8, not 0.
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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0. 14 Aug 2019, 05:03
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anuradha619
Hey, why are -2 and +2 not considered a solution for the problem?
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because they are eliminated by conditions that we apply to find solutions and to open modulus.
if we have |x| we can open modulus if we set next conditions (https://en.wikipedia.org/wiki/Absolute_value):
1. x>=0 ...
2. x<0 ...

and this conditions are stated in the solution above

If x is 0 or positive, then x^2 -2x -8 =0
--> (x-4)(x+2)=0
--> x=4 (Yes)
or
x=-2 (No!) - because x is 0 or positive

If x is negative, then x^2 +2x -8 =0
--> (x+4)(x-2)=0
--> x=-4 (Yes) or
x=2 (No!) because x is negative
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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0. 30 May 2024, 06:02
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­↧↧↧ Detailed Video Solution to the Problem ↧↧↧


We need to find the number of real solutions of \(x^2 - 2|x| - 8 = 0\)

As we have |x| in the equation so we will have two cases



As we have |x| in the equation so we will have two cases
-Case 1: x ≥ 0
=> |x| = x
=> \(x^2 - 2x - 8 = 0\)
=> \(x^2 - 4x + 2x - 8 = 0\)
=> x*(x - 4) + 2*(x - 4) = 0
=> (x - 4) * (x + 2) = 0
=> x = 4, -2


But condition was x ≥ 0
=> x = 4 is a SOLUTION		-Case 2: x < 0
=> |x| = -x
=> \(x^2 - 2(-x) - 8 = 0\)
=> \(x^2 + 2x - 8 = 0\)
=> \(x^2 + 4x - 2x - 8 = 0\)
=> x*(x + 4) - 2*(x + 4) = 0
=> (x + 4) * (x - 2) = 0
=> x = -4, 2

But condition was x < 0
=> x = -4 is a SOLUTION

=> There are 2 real solutions possible

So, Answer will be C
Hope it helps!

Watch the following video to MASTER Absolute Values

­
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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0. 04 Jun 2024, 21:25
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freedom128
x^2 -2|x| -8 =0

If x is 0 or positive, then x^2 -2x -8 =0
--> (x-4)(x+2)=0
--> x=4 (Yes) or x=-2 (No!)

If x is negative, then x^2 +2x -8 =0
--> (x+4)(x-2)=0
--> x=-4 (Yes) or x=2 (No!)

There are only two solutions:
x=-4 and x=4


Answer is (C)

Hit that +1 kudo if you like my solution

Posted from my mobile device
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­But when we take x=2 in equation 2 we see it satisfies the equation.
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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0. 05 Jun 2024, 00:26
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Apeksha2101

freedom128
x^2 -2|x| -8 =0

If x is 0 or positive, then x^2 -2x -8 =0
--> (x-4)(x+2)=0
--> x=4 (Yes) or x=-2 (No!)

If x is negative, then x^2 +2x -8 =0
--> (x+4)(x-2)=0
--> x=-4 (Yes) or x=2 (No!)

There are only two solutions:
x=-4 and x=4


Answer is (C)

Hit that +1 kudo if you like my solution

Posted from my mobile device
­But when we take x=2 in equation 2 we see it satisfies the equation.
Show more
­No, it does not.

If x = 2, then x^2 - 2|x| - 8 = 4 - 4 - 8 = -8, not 0.
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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0. 30 Nov 2025, 08:29
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